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一类KdV-Burgers方程的奇摄动解与孤子解

包立平 李瑞翔 吴立群

包立平, 李瑞翔, 吴立群. 一类KdV-Burgers方程的奇摄动解与孤子解[J]. 应用数学和力学, 2021, 42(9): 948-957. doi: 10.21656/1000-0887.420011
引用本文: 包立平, 李瑞翔, 吴立群. 一类KdV-Burgers方程的奇摄动解与孤子解[J]. 应用数学和力学, 2021, 42(9): 948-957. doi: 10.21656/1000-0887.420011
BAO Liping, LI Ruixiang, WU Liqun. Singularly Perturbed and Soliton Solutions to a Class of KdV-Burgers Equations[J]. Applied Mathematics and Mechanics, 2021, 42(9): 948-957. doi: 10.21656/1000-0887.420011
Citation: BAO Liping, LI Ruixiang, WU Liqun. Singularly Perturbed and Soliton Solutions to a Class of KdV-Burgers Equations[J]. Applied Mathematics and Mechanics, 2021, 42(9): 948-957. doi: 10.21656/1000-0887.420011

一类KdV-Burgers方程的奇摄动解与孤子解

doi: 10.21656/1000-0887.420011
基金项目: 

国家自然科学基金(51775154);浙江省重点自然科学基金(LZ15E050004)

详细信息
    作者简介:

    包立平(1962—),男,副教授,博士(E-mail: baolp@hdu.edu.cn);李瑞翔(1996—),男,硕士生(通讯作者. E-mail: 843271281@qq.com).

    通讯作者:

    李瑞翔(1996—),男,硕士生(通讯作者. E-mail: 843271281@qq.com).

  • 中图分类号: O175.29

Singularly Perturbed and Soliton Solutions to a Class of KdV-Burgers Equations

Funds: 

The National Natural Science Foundation of China(51775154)

  • 摘要: 讨论了一类具有大Reynolds数且弱频散性的KdV-Burgers方程, 在数学上表示为一类奇摄动KdV-Burgers方程.KdV-Burgers方程中含有的非线性项与频散项互补作用形成稳定向前传播的孤立子.通过数学分析, 描述了孤立子的传播途径和传播速度等物理量的发展变化规律.通过奇摄动展开方法, 构造了该问题的渐近解.首先,用Riemann-Earnshaw方法求得退化解, 得到了简单波, 该简单波波形中的任意一点与初始点都存在一个传播速度差, 这使得波在传播过程中波形不断畸变, 最终形成冲击波面, 即间断面, 在它的两侧质点的速度有一个跳跃, 且随时间不断变化;其次, 在退化解的间断曲面处做变量替换, 构造一种修正的行波变换, 得到了内解展开式的孤子解, 并证明了内外解的存在性与唯一性;最后,通过一致有界逆算子的存在性做了余项估计, 并得到渐近解的一致有效性.结果表明, KdV-Burgers方程在大Reynolds数且弱频散性的性质下,扰动集中在退化解的间断面附近,孤立子链接两侧质点,其传播途径不是时间与空间的线性形式,而是沿着退化解的间断面附近传播,形成稳定的波形.
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出版历程
  • 收稿日期:  2021-01-21
  • 修回日期:  2021-03-24

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